line segment su is a diameter of circle v. what is the measure of arc st? 56° 68° 112° 163°

Answer:

C. $112^{\circ}$

Explanation:

Step1: Recall the central - angle theorem

The measure of an arc is equal to the measure of its central angle.

Step2: Find the central angle of arc $ST$

The central angle of arc $ST$ is $\angle SVT$. Since $SU$ is a diameter, $\angle SUT = 90^{\circ}$ (angle inscribed in a semi - circle). In $\triangle SUT$, we know that the inscribed angle $\angle TSU=34^{\circ}$. The central angle $\angle SVT = 2\angle TSU$ (the central angle is twice the inscribed angle subtended by the same arc). So $\angle SVT=2\times34^{\circ}=68^{\circ}$. The measure of arc $ST$ is the measure of its central angle. But we made a wrong start above. The correct way is: The central angle of arc $ST$: The central angle corresponding to arc $ST$ is $\angle SVT$. The central angle of the whole circle is $360^{\circ}$. The central angle of arc $TU$ is also $68^{\circ}$ (because the inscribed angle $\angle TSU = 34^{\circ}$ and central angle is twice the inscribed angle subtended by the same arc). Since $SU$ is a diameter, the central angle of semi - circle $SU$ is $180^{\circ}$. The measure of arc $ST=180^{\circ}- 68^{\circ}=112^{\circ}$.